A Princeton topologist
, and one of the great expositor
s of mathematics
of the 20th century
. Several of his short volumes of lecture notes
have become the classic definitive exposition
s of their fields. These include:
Milnor and Kervaire discovered the first topological manifolds which can be given a differentiable structure (made into a smooth manifold) in several inequivalent (nondiffeomorphic) ways; they found 28 different smooth structures on the 7-sphere S7. Since then many of these manifolds have been discovered. Oddly enough, every Euclidean space Rn admits a finite number of nondiffeomorphic smooth structures (the number varying with dimension), except R4, which has a continuum of them. I don't know if there's an "explanation" per se of this fact, but it may be connected to the fact that 4-manifold topology and geometry seems to have a very rich theory full of hard problems, as compared to the lower-dimensional cases (which are simpler) and dimensions above 5 (where a group of relatively simple general principles seems to govern).